If we place the poles of a first-order system on the negative real axis using PI controller, will we in general have a step response without overshoot?

critically_damped.jl

### A Pluto.jl notebook ###
# v0.19.8

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
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# ╔═╡ 9f5d2eea-50be-4de4-94a1-7fab8006399a
using Symbolics, ControlSystems, Plots, PlutoUI

# ╔═╡ 590e213c-e081-4045-b390-d24c6ef5cf74
md"""
# Critically damped
If we place the poles of a first-order system on the negative real axis using PI controller, will we in general have a step response without overshoot?
"""

# ╔═╡ 5d745644-de82-4db8-802c-0df758f37a58
md"""
## Symbolic example
The following shows how to chose the PI parameters to place the poles with frequency ``\omega`` and damping ``\zeta``.
"""

# ╔═╡ 3f3821fc-1b1b-46c4-af40-649e2a8f8776
let
	@variables p l ω
	ζ = 1
	P = tf(1, [p, -l])
	pi_params, C = placePI(P, ω, ζ, form=:parallel)
end

# ╔═╡ 9aca5b08-8ca0-4b7b-abb4-60acc85d4f43
md"""
## Numerical example
We now turn to a numerical example. The desired bandwidth of the closed-loop system can be controlled by a slider. We can change ``l`` to be positive or negative.
"""

# ╔═╡ cbaa532f-e886-4c95-8499-04860ddd50a6
md"The pole-zero map indicates that we have a cancellation if we choose ``\omega = 1``, but not otherwise."

# ╔═╡ 27f823c8-ced5-4321-b318-16d691b0f05e
begin
	p = 1
	l = 1
	ζ = 1
	ws = @bind ω Slider(0.1:0.1:3, show_value=true, default=1)
	md"ω = $ws"
end

# ╔═╡ d95ec01a-fa37-455e-8b87-4b428b3b9267
P = tf(1, [p, l])

# ╔═╡ 9d1f3abf-7c00-47af-843e-743c6ada2fb2
pi_params, C = placePI(P, ω, ζ, form=:parallel)

# ╔═╡ b50dd9b9-bef6-492c-9c98-68907e1c75bc
T = minreal(feedback(P*C)) # closed-loop

# ╔═╡ 9701102c-2b59-4f23-97da-c43f7f27a2b5
begin
	F = tf(1, numvec(T)[])
	F = F/dcgain(F) # prefilter
end

# ╔═╡ 501e1b40-95e5-484b-b093-47da7b9c4385
pzmap(T, xlims=(-2,0.1), ylims=(-0.5, 0.5))

# ╔═╡ 0aaf4b8e-8f6f-4392-b7f6-32d30087c1a0
md"The step response and complimentary sensitivity function indicate that there will be an overshoot if we try to make the system significantly faster using feedback, i.e., if we make ``\omega > 1``."

# ╔═╡ a6f20e6d-f23c-4a73-9e0d-ddc2801b239d
plot(
    plot([
        step(T)
        step(T*F)
    ], lab=["Closed loop" "Prefiltered"]),
    bodeplot(T, plotphase=false),
)

# ╔═╡ f7a864e0-0903-4984-a605-5fff28f38337
nyquistplot(P*C, xlims=(-3,1), ylims=(-10, 0))

# ╔═╡ 7859ecf2-51c0-48fb-9a04-50f4f85f1646
md"ω = $ws"

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uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.12+3"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.ZygoteRules]]
deps = ["MacroTools"]
git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.2"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.1.0+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.41.0+1"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

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